climleltrp - Mathbox for Glauco Siliprandi

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Theorem climleltrp 45674

Description: The limit of complex number sequence 𝐹 is eventually approximated. (Contributed by Glauco Siliprandi, 26-Jun-2021.)

**Hypotheses**
Ref	Expression
climleltrp.k	⊢ Ⅎ𝑘𝜑
climleltrp.f	⊢ Ⅎ𝑘𝐹
climleltrp.z	⊢ 𝑍 = (ℤ_≥‘𝑀)
climleltrp.n	⊢ (𝜑 → 𝑁 ∈ 𝑍)
climleltrp.r	⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑘) ∈ ℝ)
climleltrp.a	⊢ (𝜑 → 𝐹 ⇝ 𝐴)
climleltrp.c	⊢ (𝜑 → 𝐶 ∈ ℝ)
climleltrp.l	⊢ (𝜑 → 𝐴 ≤ 𝐶)
climleltrp.x	⊢ (𝜑 → 𝑋 ∈ ℝ⁺)

**Assertion**
Ref	Expression
climleltrp	⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋)))

Distinct variable groups: 𝐴,𝑗,𝑘 𝑗,𝐹 𝑗,𝑁,𝑘 𝑗,𝑋,𝑘 𝑗,𝑍 𝜑,𝑗 Allowed substitution hints: 𝜑(𝑘) 𝐶(𝑗,𝑘) 𝐹(𝑘) 𝑀(𝑗,𝑘) 𝑍(𝑘)

**Proof of Theorem climleltrp**
Step	Hyp	Ref	Expression
1		climleltrp.n	. . . . 5 ⊢ (𝜑 → 𝑁 ∈ 𝑍)
2		climleltrp.z	. . . . 5 ⊢ 𝑍 = (ℤ_≥‘𝑀)
3	1, 2	eleqtrdi 2838	. . . 4 ⊢ (𝜑 → 𝑁 ∈ (ℤ_≥‘𝑀))
4		uzss 12816	. . . 4 ⊢ (𝑁 ∈ (ℤ_≥‘𝑀) → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
5	3, 4	syl 17	. . 3 ⊢ (𝜑 → (ℤ_≥‘𝑁) ⊆ (ℤ_≥‘𝑀))
6	5, 2	sseqtrrdi 3988	. 2 ⊢ (𝜑 → (ℤ_≥‘𝑁) ⊆ 𝑍)
7		climleltrp.k	. . . 4 ⊢ Ⅎ𝑘𝜑
8		climleltrp.f	. . . 4 ⊢ Ⅎ𝑘𝐹
9		uzssz 12814	. . . . 5 ⊢ (ℤ_≥‘𝑀) ⊆ ℤ
10	9, 3	sselid 3944	. . . 4 ⊢ (𝜑 → 𝑁 ∈ ℤ)
11		eqid 2729	. . . 4 ⊢ (ℤ_≥‘𝑁) = (ℤ_≥‘𝑁)
12		climleltrp.a	. . . 4 ⊢ (𝜑 → 𝐹 ⇝ 𝐴)
13		eqidd 2730	. . . 4 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑘) = (𝐹‘𝑘))
14		climleltrp.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ ℝ⁺)
15	7, 8, 10, 11, 12, 13, 14	clim2d 45671	. . 3 ⊢ (𝜑 → ∃𝑗 ∈ (ℤ_≥‘𝑁)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋))
16		nfv 1914	. . . . . 6 ⊢ Ⅎ𝑘 𝑗 ∈ (ℤ_≥‘𝑁)
17	7, 16	nfan 1899	. . . . 5 ⊢ Ⅎ𝑘(𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁))
18		simplll 774	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → 𝜑)
19		uzss 12816	. . . . . . . . . . 11 ⊢ (𝑗 ∈ (ℤ_≥‘𝑁) → (ℤ_≥‘𝑗) ⊆ (ℤ_≥‘𝑁))
20	19	ad2antlr 727	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (ℤ_≥‘𝑗) ⊆ (ℤ_≥‘𝑁))
21		simpr 484	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ (ℤ_≥‘𝑗))
22	20, 21	sseldd 3947	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → 𝑘 ∈ (ℤ_≥‘𝑁))
23	22	adantr 480	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → 𝑘 ∈ (ℤ_≥‘𝑁))
24		simpr 484	. . . . . . . 8 ⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋)
25		climleltrp.r	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑘) ∈ ℝ)
26	13, 25	eqeltrd 2828	. . . . . . . . . 10 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑘) ∈ ℝ)
27	26	adantr 480	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐹‘𝑘) ∈ ℝ)
28		climcl 15465	. . . . . . . . . . . . . . 15 ⊢ (𝐹 ⇝ 𝐴 → 𝐴 ∈ ℂ)
29	12, 28	syl 17	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ ℂ)
30	29	adantr 480	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → 𝐴 ∈ ℂ)
31	26	recnd 11202	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑘) ∈ ℂ)
32	30, 31	pncan3d 11536	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝐴 + ((𝐹‘𝑘) − 𝐴)) = (𝐹‘𝑘))
33	32	eqcomd 2735	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) → (𝐹‘𝑘) = (𝐴 + ((𝐹‘𝑘) − 𝐴)))
34	33	adantr 480	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐹‘𝑘) = (𝐴 + ((𝐹‘𝑘) − 𝐴)))
35	34, 27	eqeltrrd 2829	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐴 + ((𝐹‘𝑘) − 𝐴)) ∈ ℝ)
36		climleltrp.c	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐶 ∈ ℝ)
37	36	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → 𝐶 ∈ ℝ)
38	7, 8, 11, 10, 12, 25	climreclf 45662	. . . . . . . . . . . . . 14 ⊢ (𝜑 → 𝐴 ∈ ℝ)
39	38	ad2antrr 726	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → 𝐴 ∈ ℝ)
40	27, 39	resubcld 11606	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ((𝐹‘𝑘) − 𝐴) ∈ ℝ)
41	37, 40	readdcld 11203	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐶 + ((𝐹‘𝑘) − 𝐴)) ∈ ℝ)
42	14	rpred 12995	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝑋 ∈ ℝ)
43	42	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → 𝑋 ∈ ℝ)
44	37, 43	readdcld 11203	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐶 + 𝑋) ∈ ℝ)
45		climleltrp.l	. . . . . . . . . . . . 13 ⊢ (𝜑 → 𝐴 ≤ 𝐶)
46	45	ad2antrr 726	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → 𝐴 ≤ 𝐶)
47	39, 37, 40, 46	leadd1dd 11792	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐴 + ((𝐹‘𝑘) − 𝐴)) ≤ (𝐶 + ((𝐹‘𝑘) − 𝐴)))
48	31	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐹‘𝑘) ∈ ℂ)
49	30	adantr 480	. . . . . . . . . . . . . . 15 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → 𝐴 ∈ ℂ)
50	48, 49	subcld 11533	. . . . . . . . . . . . . 14 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ((𝐹‘𝑘) − 𝐴) ∈ ℂ)
51	50	abscld 15405	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (abs‘((𝐹‘𝑘) − 𝐴)) ∈ ℝ)
52	40	leabsd 15381	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ((𝐹‘𝑘) − 𝐴) ≤ (abs‘((𝐹‘𝑘) − 𝐴)))
53		simpr 484	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋)
54	40, 51, 43, 52, 53	lelttrd 11332	. . . . . . . . . . . 12 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ((𝐹‘𝑘) − 𝐴) < 𝑋)
55	40, 43, 37, 54	ltadd2dd 11333	. . . . . . . . . . 11 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐶 + ((𝐹‘𝑘) − 𝐴)) < (𝐶 + 𝑋))
56	35, 41, 44, 47, 55	lelttrd 11332	. . . . . . . . . 10 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐴 + ((𝐹‘𝑘) − 𝐴)) < (𝐶 + 𝑋))
57	34, 56	eqbrtrd 5129	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → (𝐹‘𝑘) < (𝐶 + 𝑋))
58	27, 57	jca 511	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑘 ∈ (ℤ_≥‘𝑁)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋)))
59	18, 23, 24, 58	syl21anc 837	. . . . . . 7 ⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋)))
60	59	adantrl 716	. . . . . 6 ⊢ ((((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) ∧ ((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋)) → ((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋)))
61	60	ex 412	. . . . 5 ⊢ (((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) ∧ 𝑘 ∈ (ℤ_≥‘𝑗)) → (((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋))))
62	17, 61	ralimdaa 3238	. . . 4 ⊢ ((𝜑 ∧ 𝑗 ∈ (ℤ_≥‘𝑁)) → (∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋))))
63	62	reximdva 3146	. . 3 ⊢ (𝜑 → (∃𝑗 ∈ (ℤ_≥‘𝑁)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℂ ∧ (abs‘((𝐹‘𝑘) − 𝐴)) < 𝑋) → ∃𝑗 ∈ (ℤ_≥‘𝑁)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋))))
64	15, 63	mpd 15	. 2 ⊢ (𝜑 → ∃𝑗 ∈ (ℤ_≥‘𝑁)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋)))
65		ssrexv 4016	. 2 ⊢ ((ℤ_≥‘𝑁) ⊆ 𝑍 → (∃𝑗 ∈ (ℤ_≥‘𝑁)∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋)) → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋))))
66	6, 64, 65	sylc 65	1 ⊢ (𝜑 → ∃𝑗 ∈ 𝑍 ∀𝑘 ∈ (ℤ_≥‘𝑗)((𝐹‘𝑘) ∈ ℝ ∧ (𝐹‘𝑘) < (𝐶 + 𝑋)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 Ⅎwnf 1783 ∈ wcel 2109 Ⅎwnfc 2876 ∀wral 3044 ∃wrex 3053 ⊆ wss 3914 class class class wbr 5107 ‘cfv 6511 (class class class)co 7387 ℂcc 11066 ℝcr 11067 + caddc 11071 < clt 11208 ≤ cle 11209 − cmin 11405 ℤcz 12529 ℤ_≥cuz 12793 ℝ⁺crp 12951 abscabs 15200 ⇝ cli 15450

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2701 ax-rep 5234 ax-sep 5251 ax-nul 5261 ax-pow 5320 ax-pr 5387 ax-un 7711 ax-cnex 11124 ax-resscn 11125 ax-1cn 11126 ax-icn 11127 ax-addcl 11128 ax-addrcl 11129 ax-mulcl 11130 ax-mulrcl 11131 ax-mulcom 11132 ax-addass 11133 ax-mulass 11134 ax-distr 11135 ax-i2m1 11136 ax-1ne0 11137 ax-1rid 11138 ax-rnegex 11139 ax-rrecex 11140 ax-cnre 11141 ax-pre-lttri 11142 ax-pre-lttrn 11143 ax-pre-ltadd 11144 ax-pre-mulgt0 11145 ax-pre-sup 11146

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2533 df-eu 2562 df-clab 2708 df-cleq 2721 df-clel 2803 df-nfc 2878 df-ne 2926 df-nel 3030 df-ral 3045 df-rex 3054 df-rmo 3354 df-reu 3355 df-rab 3406 df-v 3449 df-sbc 3754 df-csb 3863 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3934 df-nul 4297 df-if 4489 df-pw 4565 df-sn 4590 df-pr 4592 df-op 4596 df-uni 4872 df-iun 4957 df-br 5108 df-opab 5170 df-mpt 5189 df-tr 5215 df-id 5533 df-eprel 5538 df-po 5546 df-so 5547 df-fr 5591 df-we 5593 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-pred 6274 df-ord 6335 df-on 6336 df-lim 6337 df-suc 6338 df-iota 6464 df-fun 6513 df-fn 6514 df-f 6515 df-f1 6516 df-fo 6517 df-f1o 6518 df-fv 6519 df-riota 7344 df-ov 7390 df-oprab 7391 df-mpo 7392 df-om 7843 df-2nd 7969 df-frecs 8260 df-wrecs 8291 df-recs 8340 df-rdg 8378 df-er 8671 df-pm 8802 df-en 8919 df-dom 8920 df-sdom 8921 df-sup 9393 df-inf 9394 df-pnf 11210 df-mnf 11211 df-xr 11212 df-ltxr 11213 df-le 11214 df-sub 11407 df-neg 11408 df-div 11836 df-nn 12187 df-2 12249 df-3 12250 df-n0 12443 df-z 12530 df-uz 12794 df-rp 12952 df-fl 13754 df-seq 13967 df-exp 14027 df-cj 15065 df-re 15066 df-im 15067 df-sqrt 15201 df-abs 15202 df-clim 15454 df-rlim 15455

This theorem is referenced by: smflimlem2 46770

W3C validator